The (𝐺^′/𝐺)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

Abstract

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the (𝐺^′/𝐺)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (𝐺^′/𝐺)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

1. Introduction

The investigation of exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) plays an important role in the analysis of complex physical phenomena. The NLPDEs appear in physical sciences, various scientific and engineering problems, such as, fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, chemistry and many others. In recent years, to obtain exact traveling wave solutions of NLPDEs, many effective and powerful methods have been presented in the literature, such as the Backlund transformation [1], the tanh function method [2], the extended tanh function method [3], the variational iteration method [4], the Adomian decomposition method [5, 6], the homotopy perturbation method [7], the F-expansion method [8], the Hirota’s bilinear method [9], the exp-function method [10], the Cole-Hopf transformation [11], the general projective Riccati equation method [12] and others [13–20], Nowaday, searching analytical solutions of the NLPDEs has become more lucrative partly due to the accessibility computer symbolic systems, like Maple, Mathematica, and Matlab, which help us to calculate the complicated and wearisome algebraic calculations on computer.

Recently, Wang et al. [21] introduced a new direct method called the (𝐺^′/𝐺)-expansion method to seek traveling wave solutions of the nonlinear evolution equations. Abazari [22] implemented the (𝐺^′/𝐺)-expansion method to NLEEs related to fluid mechanics. Zheng [23] used the method for getting exact traveling wave solutions of two different types of equations. Feng et al. [24] used the method to seek solutions of the Kolmogorov-Petrovskii-Piskunov equation. Liu et al. [25] concerned the method to simplified MCH equation and Gardner equation. Feng [26] applied the method to the seventh-order Sawada-Kotera equation.

At present time, Guo and Zhou [27] first proposed the extended (𝐺^′/𝐺)-expansion method based on new ansatz. They applied the method to the Whitham-Broer-Kaup-Like equations and couple Hirota-Satsuma KdV equations. Zayed et al. [28] used extended (𝐺^′/𝐺)-expansion method to some nonlinear PDEs in mathematical physics. Zhang et al. [29] presented an improved (𝐺^′/𝐺)-expansion method for solving nonlinear evolution equations (NLEEs). Li et al. [30] introduced (𝐺^′/𝐺,1/𝐺)-expansion method to obtain traveling wave solutions of the Zakharov equations. Hayek [31] proposed extended (𝐺^′/𝐺)-expansion method for constructing exact solutions to the KdV Burgers equations with power-law nonlinearity.

In this paper, our aim is to investigate abundant new exact traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon equation by the (𝐺^′/𝐺)-expansion method.

2. Description of the (𝐺^′/𝐺)-Expansion Method

Suppose that the nonlinear partial differential equation is of the form 𝑃𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑥𝑡,𝑢_𝑥𝑥,...=0, (2.1) where 𝑢=𝑢(𝑥,𝑡) is an unknown function and 𝑃is a polynomial in 𝑢=𝑢(𝑥,𝑡) which has various partial derivatives, and the highest order derivatives and nonlinear terms are involved.

The main steps of the (𝐺^′/𝐺)-expansion method [21] are conveyed in the following.

Step 1. The traveling wave transformation: 𝑢(𝑥,𝑡)=𝑣(𝜉),𝜉=𝑥+𝑠𝑡, (2.2) where 𝑠 is the wave speed, 𝜉 is the combination of two independent variables 𝑥 and 𝑡, transform (2.1) into an ordinary differential equation for 𝑣(𝜉):𝑄𝑣,𝑠𝑣^′,𝑣^′,𝑠²𝑣^′′,𝑠𝑣^′′,𝑣^′′,...=0, (2.3) where primes denote the ordinary derivatives with respect to𝜉.

Step 2. If possible, integrate (2.3) term by term for one or more times, yields constant(s) of integration. For simplicity, the integration constant(s) may be zero.

Step 3. Suppose that the solution of (2.3) can be expressed by a polynomial in (𝐺′/𝐺): 𝑣(𝜉)=𝛼_𝑚𝐺^′𝐺^𝑚+𝛼_𝑚−1𝐺^′𝐺^𝑚−1+⋯+𝛼₀, (2.4) where 𝛼_𝑛 (𝑛=0,1,2,3,…,𝑚) are constants and 𝛼_𝑚≠0, and 𝐺=𝐺(𝜉) satisfies the second order linear ordinary differential equation (LODE): 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (2.5) where 𝜆 and 𝜇 are arbitrary constants. The positive integer 𝑚 can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (2.3).

Step 4. Substituting (2.4) together with (2.5) into (2.3) yields an algebraic equation involving powers of (𝐺′/𝐺). Then, equating coefficients of each power of (𝐺′/𝐺) to zero yields a set of algebraic equations for 𝛼_𝑛 (𝑛=0,1,2,3,…,𝑚), 𝑠, 𝜆, and 𝜇. Since the general solution of (2.5) is known for us, then substituting 𝛼_𝑛 (𝑛=0,1,2,3,…,𝑚) and 𝑠 together with general solution of (2.5) into (2.4), we obtain exact traveling wave solutions of the nonlinear partial differential equation (2.1).

3. Application of the Method

In this section, we apply the method to construct the hyperbolic, the trigonometric, and the rational function solutions of the fifth-order Caudrey-Dodd-Gibbon equation, and the solutions are shown in graphs.

3.1. The Caudrey-Dodd-Gibbon Equation

We consider the fifth-order Caudrey-Dodd-Gibbon equation: 𝑢_𝑡+𝑢_{𝑥𝑥𝑥𝑥𝑥}+30𝑢𝑢_𝑥𝑥𝑥+30𝑢_𝑥𝑢_𝑥𝑥+180𝑢²𝑢_𝑥=0. (3.1) Now, we use the transformation equation (2.2) into (3.1), which yields, 𝑠𝑣^′+𝑣⁽⁵⁾+30𝑣𝑣^′′′+30𝑣^′𝑣^′′+180𝑣²𝑣^′=0, (3.2) where primes denote the derivative with respect to 𝜉.

Equation. (3.2) is integrable, therefore, integrating once with respect to 𝜉 yields 𝑠𝑣+𝑣⁽⁴⁾+30𝑣𝑣^′′+60𝑣³+𝐶=0, (3.3) where 𝐶 is an integral constant, that is, to be determined later.

Considering the homogeneous balance between 𝑣⁽⁴⁾ and 𝑣³ in (3.3), we obtain 𝑚=2. Therefore, the solution of (3.3) is of the form: 𝑣(𝜉)=𝛼₂𝐺^′𝐺²+𝛼₁𝐺^′𝐺+𝛼₀, (3.4) where 𝛼₂, 𝛼₁, and 𝛼₀ are constants and 𝛼₂≠0. Substituting (3.4) together with (2.5) into (3.3) and collecting all terms with the same power of (𝐺′/𝐺) and setting each coefficient of the resulted polynomial to zero, we obtain a set of algebraic equations for 𝛼₂, 𝛼₁, 𝛼₀, 𝑠, 𝜆, 𝜇, and 𝐶 as follows: 60𝛼³₂+120𝛼₂+180𝛼²₂=0,180𝛼²₂𝛼₁+240𝛼₂𝛼₁+300𝛼²₂𝜆+336𝛼₂𝜆+24𝛼₁=0,180𝛼₂𝛼²₁+180𝛼²₂𝛼₀+240𝛼²₂𝜇+120𝛼²₂𝜆²+180𝛼₀𝛼₂+330𝛼₂𝜆²+240𝛼₂𝜇+60𝛼₁𝜆+60𝛼²₁+390𝛼₂𝛼₁𝜆=0,360𝛼₂𝛼₁𝛼₀+60𝛼³₁+440𝛼₂𝜆𝜇+90𝛼²₁𝜆+60𝛼₀𝛼₁+40𝛼₁𝜇+130𝛼₂𝜆³+50𝛼₁𝜆²+180𝛼²₂𝜆𝜇+300𝛼₂𝛼₁𝜇+150𝛼₂𝛼₁𝜆²+300𝛼₀𝛼₂𝜆=0,180𝛼₂𝛼²₀+180𝛼²₁𝛼₀+210𝛼₂𝛼₁𝜆𝜇+60𝛼²₂𝜇²+60𝛼²₁𝜇+30𝛼²₁𝜆²+𝑠𝛼₂+136𝛼₂𝜇²+16𝛼₂𝜆⁴+15𝛼₁𝜆³+60𝛼₁𝜆𝜇+232𝛼₂𝜆²𝜇+240𝛼₀𝛼₂𝜇+90𝛼₀𝛼₁𝜆+120𝛼₀𝛼₂𝜆²=0,180𝛼₁𝛼²₀+180𝛼₀𝛼₂𝜆𝜇+16𝛼₁𝜇²+𝑠𝛼₁+𝛼₁𝜆⁴+22𝛼₁𝜆²𝜇+120𝛼₂𝜆𝜇²+30𝛼₂𝜆³𝜇+60𝛼₁𝛼₂𝜇²+30𝛼²₁𝜆𝜇+60𝛼₀𝛼₁𝜇+30𝛼₀𝛼₁𝜆²=0,60𝛼³₀+14𝛼₂𝜆²𝜇²+𝛼₁𝜆³𝜇+8𝛼₁𝜆𝜇²+16𝛼₂𝜇³+𝐶+60𝛼₀𝛼₂𝜇²+𝑠𝛼₀+30𝛼₀𝛼₁𝜆𝜇=0. (3.5) Solving the system of algebraic equations with the aid of Maple 13, we obtain two different sets of solution.

Case 1. One has 𝛼₀=−16𝜆²−43𝜇,𝛼₁=−2𝜆,𝛼₂=−2,𝑠=−𝜆⁴+8𝜇𝜆²−16𝜇²,1𝐶=9𝜆⁶−43𝜆⁴𝜇+163𝜆²𝜇²−649𝜇³, (3.6) where 𝜆 and 𝜇 are arbitrary constants.

Case 2. One has 𝛼₁=−𝜆,𝛼₂=−1,𝑠=−180𝛼²₀−30𝜆²𝛼₀−22𝜇𝜆²−76𝜇²−𝜆⁴−240𝜇𝛼₀,𝐶=120𝛼³₀+22𝜆²𝜇²+𝜆⁴𝜇+16𝜇³+136𝛼₀𝜇²+30𝜆²𝛼²₀+52𝛼₀𝜆²𝜇+𝛼₀𝜆⁴+240𝜇𝛼²₀, (3.7) where 𝛼₀, 𝜆, and 𝜇 are arbitrary constants.

Case 1. Substituting (3.6) into (3.4) yields 𝑣(𝜉)𝐺=−2^′𝐺²𝐺−2𝜆^′𝐺−16𝜆²−43𝜇. (3.8)

Substituting the general solution of (2.5) into (3.8), we obtain three types of traveling wave solutions of (3.3) as follows.

Hyperbolic Function Solutions
When 𝜆²−4𝜇>0, substituting the general solution of (2.5) into (3.8), we obtain the following traveling wave solution of (3.3): 𝑣(𝜉)𝜆=−²−4𝜇2𝐶₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉²+13𝜆²−43𝜇, (3.9) where 𝜉=𝑥+(−𝜆⁴+8𝜇𝜆²−16𝜇²)𝑡, 𝐶₁, and 𝐶₂ are arbitrary constants. The various known results can be rediscovered, if 𝐶₁ and 𝐶₂ are taken as special values. For example:

(i)if 𝐶₁=0 but 𝐶₂≠0,we obtain 𝑣(𝜉)𝜆=−²−4𝜇2coth²12𝜆²1−4𝜇𝜉+3𝜆²−43𝜇. (3.10)(ii)If 𝐶₂=0 but 𝐶₁≠0, we obtain 𝑣(𝜉)𝜆=−²−4𝜇2tanh²12𝜆²1−4𝜇𝜉+3𝜆²−43𝜇. (3.11)(iii)If 𝐶₁≠0and 𝐶²₁>𝐶²₂, we obtain 𝑣(𝜉)=𝜆²−4𝜇2sech²12𝜆²−4𝜇𝜉+𝜉₀−𝜆²6+23𝜇, (3.12) where 𝜉₀=tanh⁻¹(C₂/C₁).

Trigonometric Function Solutions
When 𝜆²−4𝜇<0, we obtain 𝑣(𝜉)=−4𝜇−𝜆²2−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉²+13𝜆²−43𝜇, (3.13) where 𝜉=𝑥+(−𝜆⁴+8𝜇𝜆²−16𝜇²)𝑡, 𝐶₁ and 𝐶₂ are arbitrary constants.

The various known results can be rediscovered, if 𝐶₁ and 𝐶₂ are taken as special values.

Rational Function Solution
When 𝜆²−4𝜇=0,we obtain 𝑣(𝜉)=−2𝐶²₂𝐶₁+𝐶₂𝜉²+13𝜆²−43𝜇, (3.14) where 𝜉=𝑥+(−𝜆⁴+8𝜇𝜆²−16𝜇²)𝑡, 𝐶₁, and 𝐶₂ are arbitrary constants.

Case 2. Substituting (3.7) into (3.4) yields 𝑣(𝜉)𝐺=−^′𝐺²𝐺−𝜆^′𝐺+𝛼₀. (3.15)

Substituting the general solution of (2.5) into (3.15), we obtain three types of traveling wave solutions of (3.3) as follows.

Hyperbolic Function Solutions
When 𝜆²−4𝜇>0, we obtain 𝑣(𝜉)𝜆=−²−4𝜇2𝐶₁sinh(1/2)𝜆²−4𝜇𝜉+𝐶₂cosh(1/2)𝜆²−4𝜇𝜉𝐶₁cosh(1/2)𝜆²−4𝜇𝜉+𝐶₂sinh(1/2)𝜆²−4𝜇𝜉²+14𝜆²+𝛼₀, (3.16) where 𝜉=𝑥+(−180𝛼²₀−30𝜆²𝛼₀−22𝜇𝜆²−76𝜇²−𝜆⁴−240𝜇𝛼₀)𝑡. 𝐶₁ and 𝐶₂ are arbitrary constants.

The various known results can be rediscovered, if 𝐶₁ and 𝐶₂ are taken as special values.

Trigonometric Function Solutions
When 𝜆²−4𝜇<0, we obtain 𝑣(𝜉)=−4𝜇−𝜆²4−𝐶₁sin(1/2)4𝜇−𝜆²𝜉+𝐶₂cos(1/2)4𝜇−𝜆²𝜉𝐶₁cos(1/2)4𝜇−𝜆²𝜉+𝐶₂sin(1/2)4𝜇−𝜆²𝜉²+14𝜆²+𝛼₀, (3.17) where 𝜉=𝑥+(−180𝛼²₀−30𝜆²𝛼₀−22𝜇𝜆²−76𝜇²−𝜆⁴−240𝜇𝛼₀)𝑡.𝐶₁ and 𝐶₂ are arbitrary constants. The various known results can be rediscovered, if 𝐶₁ and 𝐶₂ are taken as special values. For example:

(i)if 𝐶₁=0 but 𝐶₂≠0, we obtain 𝑣(𝜉)=−4𝜇−𝜆²4cot²124𝜇−𝜆²1𝜉+4𝜆²+𝛼₀. (3.18)(ii)If 𝐶₂=0 but 𝐶₁≠0,we obtain 𝑣(𝜉)=𝜆²−4𝜇4tan²124𝜇−𝜆²1𝜉+4𝜆²+𝛼₀. (3.19)(iii)If 𝐶₁≠0 and 𝐶²₁>𝐶²₂, we obtain 𝑣(𝜉)=−4𝜇−𝜆²4cot²124𝜇−𝜆²𝜉+𝜉₀+14𝜆²+𝛼₀, (3.20) where 𝜉₀=tan⁻¹𝐶₂/𝐶₁.

Rational Function Solution
When 𝜆²−4𝜇=0, 𝑣(𝜉)𝐶=−²²𝐶₁+𝐶₂𝜉²+14𝜆²+𝛼₀, (3.21) where 𝜉=𝑥+(−180𝛼²₀−30𝜆²𝛼₀−22𝜇𝜆²−76𝜇²−𝜆⁴−240𝜇𝛼₀)𝑡. 𝐶₁ and 𝐶₂ are arbitrary constants.

3.2. Discussion

The solutions of the CDG (3.1) are investigated by different methods, such as Jin [32] investigated solutions by the variational iteration method, Salas [33] by the projective Riccati equation method, and Wazwaz [34] by using the tanh method. To the best of our awareness the CDG equation is not solved by the prominent (𝐺′/𝐺)-expansion method. In this paper, we solve this equation by the (𝐺′/𝐺)-expansion method. It is noteworthy to point out that our attained solutions are new and cannot be found from above author’s solutions by any choice of arbitrary constants.

3.3. Graphical Representations of the Solutions

The solutions are shown in the graphs with the aid of Maple 13 in Figures 1–5.

Figure 1 

Solitary wave obtained from solution (3.10), for 𝜆=3, 𝜇=2.

Figure 2 

Solitary wave obtained from solution (3.11), for 𝜆=3, 𝜇=2.

Figure 3 

Solitary wave obtained from solution (3.14), for 𝜆=2, 𝜇=1, 𝐶1=1, 𝐶2=2.

Figure 4 

Solitary wave obtained from solution (3.19), for 𝜆=1, 𝜇=1, 𝛼0=1.

Figure 5 

Solitary wave obtained from solution (3.19), for 𝜆=1, 𝜇=1, 𝛼0=1.

4. Conclusions

In this paper, three types of traveling wave solutions, such as the hyperbolic, the trigonometric, and the rational functions of the Caudrey-Dodd-Gibbon equation are successfully obtained by using the (𝐺′/𝐺)-expansion method. Exact traveling wave solutions of this equation have many potential applications in engineering and mathematical physics. The obtained solutions also show that the method is effective, more powerful, and simple for searching exact traveling wave solutions of the NLPDEs. The method can be applied in different types of NLEEs and it is our task in the future.

Acknowledgments

This paper is supported by USM short term Grant (no. 304/PMATHS/6310072). The authors would like to thank the School of Mathematical Sciences, USM, for providing related research facilities.